Rich Audu hit a pair of key 3-pointers in the second half for the Saints (4-13, 2-4), who had lost 11 in a row before defeating Canisius 57-54 Friday night.
Saint Peter's (6-10, 1-5) fell behind by 13 points in the first half but trailed just 49-47 with 5:07 remaining.
Siena ran off the next eight points, including consecutive treys by Audu, who played the rest of the way after Ryan Oliver injured his knee in the game's second minute.
The Saints had a huge edge in free throws, making 27 of 36, while Saint Peter's made 5 of 12. Anosike, who was averaging 45.8 percent from the line, made 12 of 17.
Desi Washington led the Peacocks with 16 points.
Leaving early for the NBA is a risky decision. Some players -- Skal Labissiere, Dejounte Murray -- slid in the draft. Some -- Isaiah Taylor, Troy Williams -- weren't drafted at all.
College basketball coaches across the country will hit the recruiting trail hard in July to target the Class of 2017's top prospects. Which players are the nation's top programs targeting?
The Tar Heels could lose Justin Jackson and Kennedy Meeks after next season. Coach Roy Williams has nearly every top-10 2017 prospect on his board.